I have seen in many references that the field equations for a quantum field in curved space time is mentioned to be 'manifestly co-variant'. The procedure that has been followed to arrive at these field equations roughly includes the following steps(For a scalar field):

In the classical action for the scalar field one replaces the minkowski metric with a general metric $g_{\nu\mu}$, and then the covariant volume element $\sqrt{-g} d^4x$ is used for the integration. Extremizing this action would give the classical generally co-variant field equations. Then one performs mode expansion for the solution of the above field equation and does the quantisation by introducing appropriate operators. The resulting expression for the field operators are often termed as being manifestly generally co-variant.

The question: How different is the meaning of 'manifestly generally co-variant' from the general co-variance used in general relativity? If they mean the same thing,it appears that it would be fair to call quantum field theory in curved space-time to be general relativistic quantum field theory. Physically this seems to suggest that the equations of motion describing the quantum fields in curved space time retain their form under arbitrary coordinate transformation. Is this sensible?


This is exactly what is meant -- the theory you write down will be completely independent of the coordinates which you use to describe it. You don't even need to be in curved spacetime to use this -- simply write flat space in funky coordinates and you'll be using this formulation.

Your statement that one could call this a "general relativitic quantum field theory" is not quite accurate, however. The point of general relativity is that the metric tensor is a dynamical variable satisfying Einstein's equations. However, writing the action for a field living on a fixed background does not give any dynamics to the metric tensor, and thus has nothing to do with general relativity.

  • $\begingroup$ So, the formalism just allows us to express the field equations in a coordinate independent way like any other tonsorial equation. I suppose then that, the formalism is too general and therefore does not incorporate both the weak equivalence principle, stated as ''The local effects of motion in a curved spacetime (gravitation) are indistinguishable from those of an accelerated observer in flat spacetime, without exception''(From wikipedia) and the strong equivalence principle. $\endgroup$ – GlaDos Jun 26 at 21:46

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